6 POSITIVE LINEAR FORMS, POSITIVE HUBERT FORMS 355 element y of C, we have xO*) = xCvX and there exists a character % such that p(x*x) = X(x*x). If g is the restriction of % to C, we have \g(y)\ ^p(y) for all y E C. Now use Section 15.4, Problem 18(e); Section 12.15, Problem 4; and Problem 2(c) above). (b) Conversely, if for each .x e A there exists a positive linear form /on A such that /O) = 1 and/O**)1/2 = p(x\ then A is hermitian. (If a = p(x*x) and u ae x*x, show that /(#2) ^ a2 for all positive linear forms / on A such that f(e) 1, and deduce from the hypothesis that p(u2) ^ a2, so that p(oc.e x*x) <i a ; then use Section 15.4, Problem 19.) (c) Suppose that A is hermitian, and let S(A) be the subspace of the dual A' of the Banach space A, consisting of the positive linear forms / on A such that f(e) = 1 . Then S(A) is compact and metrizable for the weak topology (12.15.9). Let (/) be a dense sequence in S(A), and let $*-* Ģ/(?) be the representation of A on a separable Hilbert space Hn constructed from / by the procedure of (15.6.10). We obtain a representation U of A on the Hilbert space H which is the Hilbert sum of the Hn (6.4) by putting U(s) Z *ŧ = Z WJ) " *Ŧ Show that II u(s) II = XA and hence that the kernel of U is the radical 91 of A, and U is continuous. (d) A star algebra A with unit element is hermitian (1 5.4.1 2), and for each x e A we have \\x\\ p(x) (Section 15.4, Problem 4(a)). If A is separable, deduce that there exists an isometric isomorphism U of A onto a closed involutive subalgebra of JS?(E) (where E is a separable Hilbert space), such that U(s*) = (U(s))* for all s e A. (e) If E is a separable Hilbert space, the algebra A = C 1E Đ -^(E) is a hermitian Banach algebra relative to the norm ||Ģ 1E + w||2 = |f | H- \\u\\z, in the notation of (15.4.8). Compare ||w||2 and p(u) for u e ^f2(E). 4. Let A be a Banach algebra with unit element, and let #i->jc* be a (not necessarily continuous) involution on A. Show that the following properties are equivalent: (a) There exists a norm on A, equivalent to the given norm, relative to which A is a star algebra. (j3) There exists a number A> 0 such that ||jc*jt|| S> A||**|| - \\x\\ for all x e A. (y) The set of unitary elements of A is bounded. (3) The involution XH>X* is hermitian, and there exists a number ju,>0 such that ||a|| ^ /x p(a) for all hermitian elements a e A. (To show that (8) implies (a), show that (8) implies that \\x\\ <^ 2fjup(x)t by writing x = a + ib with a, b hermitian, and using Section 15.4, Problem 18(d) and 18(g). To show that (]8) implies (8), show first that if a is hermitian, then A||fl|| ^p(a\ by using (15.2.7); deduce that if x is normal, then A2||xŧ||- \\(x*?\\ <p(x*x)n for all integers n > 1, and use Section 15.4, Problem 19 to show that A is hermitian. Finally, to show that (y) implies (S), observe that for each hermitian element a such that p(a) < I there exists in A a hermitian element 6, commuting with a, such that b2 = e a2 (Section 15.4, Problem 17); consequently u a + ib is unitary, and a = i(Ŧ H- Ŧ*).) 5. Let A be a Banach algebra with unit element e, endowed with a hermitian involution x*-*x*. (a) Let xeA be such that p(x)< 1. Show that x(e-~x*x)1'2 Ŧ (*- xx*)1/2x (cf. Section 15.2, Problem 11(0). (b) Under the same hypotheses, show that the function = 0? - xx*)-ll\Ģe + x)(e + Ģx*r1(e - x*x)li*ntain a neighborhood of e in H, which would